To study the impact of spatial heterogeneity of environment and the movement of individuals on the persistence and extinction of a disease, we propose a spatial SIR reaction diffusion model. First, the basic reproduction number $R_{0}$ is defined and the effects of the diffusion of infected individuals on $R_{0}$ are analyzed. It is shown that if $R_{0}<1$, the disease-free equilibrium is globally asymptotically stable. If $R_{0}>1$, the disease-free equilibrium is unstable. Second, the existence and stability of endemic equilibrium are studied by the bifurcation theory for low risk domains. The results show that reducing the diffusion of the infected individuals is not the optimal strategy of eradicating diseases. But the instability of the endemic equilibrium implies that the disease can be controlled eventually.